Harary [1]} called this construction the composition. A lexicographic product \({G}[{H}]\)  can be regarded to have \(|V(G)|\)  pseudo-vertices. Each of them is isomorphic to \(H\)  and two pseudo-vertices are “adjacent” if the corresponding vertices of \(G\)  are adjacent. Graph invariants of lexicographic product have been investigated by, for example, Geller and Stahl [2]}. Independence complexes of lexicographic products are studied by Vander Meulen and Van Tuyl [3]} from combinatorial point of view.
